Finite-time singularity formation in Hele-Shaw systems

We investigate the behavior of the interface between two fluids in a two-dimensional flow driven by surface tension. The geometry is chosen so that one can apply a variant of the lubrication approximation and so that the more-viscous fluid will have a tendency to change its topology by separating into two masses. Simulations are used to show that, with appropriate initial and boundary conditions, this separation can occur in a finite time. We particularly focus our attention at the pinch point, i.e., the space-time point at which the width of the viscous fluid first goes to zero. The lubrication approximation used contains a parameter ρ which measures the strength of the inertial forces. Since the fluid velocity diverges as the pinch is approached, the behavior is qualitatively different for small ρ and for ρ=0. Simulations and asymptotic analyses are used to isolate this difference. For ρ=0, at the pinch time there is a region of space in which the width grows quadratically as one moves away from the pinch. The curvatures, however, are different on the two sides of the pinch. In contrast, when ρ is different from zero, the width rises nearly linearly with distance from the pinch.